"Fermat-Pythagorean" Theorem

Number Theory Level 3

How many tuples of positive integers a ≤ b ≤ c ≤ d a \leq b \leq c \leq d satisfying gcd ⁡ ( a , b , c , d ) = 1 \gcd(a,b,c,d) = 1 are there, such that

a 3 + b 3 + c 3 = d 3 ? a^3 + b^3 + c^3 = d ^3 ?

Infinitely many 0 Finite and more than 1000 Between 1 and 1000

Chung Kevin by Chung Kevin , 46, Australia

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2 solutions

Chaebum Sheen
Oct 20, 2016

It is easy to show that there are infinite such tuples, and a a can even be any positive integer.

The following equation is well known:

n 3 + ( 3 n 2 + 2 n + 1 ) 3 + ( 3 n 3 + 3 n 2 + 2 n ) 3 = ( 3 n 3 + 3 n 2 + 2 n + 1 ) 3 n^3+(3n^2+2n+1)^3+(3n^3+3n^2+2n)^3=(3n^3+3n^2+2n+1)^3

See here for more.

3 Helpful 1 Interesting 0 Brilliant 0 Confused

That's the equation that I've seen. It's somewhat surprising!

Chung Kevin - 4 years, 7 months ago
Kaushik Chandra
Oct 8, 2016

So guys u all know that the number line never ends and we can take any sort of number either too large or too small. We have the relationship a^3+b^3+c^3= d^3 where they can be satisfied for infinity values as the number line never ends........

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Note that these variables must be integers. For example, there is no integer solution to 1 3 + 2 3 + 3 3 = d 3 1 ^3 + 2 ^ 3 + 3 ^ 3 = d ^ 3 .

The "smallest" solution occurs at 3 3 + 4 4 + 5 3 = 6 3 3 ^3 + 4^4 + 5^3 = 6 ^ 3 . Because I asked for "coprime" integers, we cannot just multiply throughout by k k .

Chung Kevin - 4 years, 8 months ago

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